from sage.all import *
+def project_span(K, K2 = None):
+ r"""
+ Return a "copy" of ``K`` embeded in a lower-dimensional space.
+
+ By default, we will project ``K`` into the subspace spanned by its
+ rays. However, if ``K2`` is not ``None``, we will project into the
+ space spanned by the rays of ``K2`` instead.
+
+ EXAMPLES::
+
+ sage: K = Cone([(1,0,0), (0,1,0)])
+ sage: project_span(K)
+ 2-d cone in 2-d lattice N
+ sage: project_span(K).rays()
+ N(1, 0),
+ N(0, 1)
+ in 2-d lattice N
+
+ sage: K = Cone([(1,0,0), (0,1,0)])
+ sage: K2 = Cone([(0,1)])
+ sage: project_span(K, K2).rays()
+ N(1)
+ in 1-d lattice N
+
+ """
+ # Allow us to use a second cone to generate the subspace into
+ # which we're "projecting."
+ if K2 is None:
+ K2 = K
+
+ # Use these to generate the new cone.
+ cs1 = K.rays().matrix().columns()
+
+ # And use these to figure out which indices to drop.
+ cs2 = K2.rays().matrix().columns()
+
+ perp_idxs = []
+
+ for idx in range(0, len(cs2)):
+ if cs2[idx].is_zero():
+ perp_idxs.append(idx)
+
+ solid_cols = [ cs1[idx] for idx in range(0,len(cs1))
+ if not idx in perp_idxs
+ and not idx >= len(cs2) ]
+
+ m = matrix(solid_cols)
+ L = ToricLattice(len(m.rows()))
+ J = Cone(m.transpose(), lattice=L)
+ return J
+
+
def discrete_complementarity_set(K):
r"""
Compute the discrete complementarity set of this cone.
OUTPUT:
- A ``MatrixSpace`` object `M` such that every matrix `L \in M` is
- Lyapunov-like on this cone.
+ A list of matrices forming a basis for the space of all
+ Lyapunov-like transformations on the given cone.
+
+ EXAMPLES:
+
+ The trivial cone has no Lyapunov-like transformations::
+
+ sage: L = ToricLattice(0)
+ sage: K = Cone([], lattice=L)
+ sage: LL(K)
+ []
+
+ The Lyapunov-like transformations on the nonnegative orthant are
+ simply diagonal matrices::
+
+ sage: K = Cone([(1,)])
+ sage: LL(K)
+ [[1]]
+
+ sage: K = Cone([(1,0),(0,1)])
+ sage: LL(K)
+ [
+ [1 0] [0 0]
+ [0 0], [0 1]
+ ]
+
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: LL(K)
+ [
+ [1 0 0] [0 0 0] [0 0 0]
+ [0 0 0] [0 1 0] [0 0 0]
+ [0 0 0], [0 0 0], [0 0 1]
+ ]
+
+ Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
+ `L^{3}_{\infty}` cones [Rudolf et al.]_::
+
+ sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
+ sage: LL(L31)
+ [
+ [1 0 0]
+ [0 1 0]
+ [0 0 1]
+ ]
+
+ sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
+ sage: LL(L3infty)
+ [
+ [1 0 0]
+ [0 1 0]
+ [0 0 1]
+ ]
+
+ TESTS:
+
+ The inner product `\left< L\left(x\right), s \right>` is zero for
+ every pair `\left( x,s \right)` in the discrete complementarity set
+ of the cone::
+
+ sage: K = random_cone(max_dim=8, max_rays=10)
+ sage: C_of_K = discrete_complementarity_set(K)
+ sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
+ sage: sum(map(abs, l))
+ 0
"""
V = K.lattice().vector_space()
An integer representing the Lyapunov rank of the cone. If the
dimension of the ambient vector space is `n`, then the Lyapunov rank
will be between `1` and `n` inclusive; however a rank of `n-1` is
- not possible (see the first reference).
+ not possible for any cone.
.. note::
cone and Lyapunov-like transformations, Mathematical Programming, 147
(2014) 155-170.
+ .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
+ Improper Cone. Work in-progress.
+
.. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
optimality constraints for the cone of positive polynomials,
Mathematical Programming, Series B, 129 (2011) 5-31.
sage: b == n-1
False
+ In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
+ Lyapunov rank `n-1` in `n` dimensions::
+
+ sage: K = random_cone(max_dim=10, max_rays=16)
+ sage: b = lyapunov_rank(K)
+ sage: n = K.lattice_dim()
+ sage: b == n-1
+ False
+
+ The calculation of the Lyapunov rank of an improper cone can be
+ reduced to that of a proper cone [Orlitzky/Gowda]_::
+
+ sage: K = random_cone(max_dim=15, max_rays=25)
+ sage: actual = lyapunov_rank(K)
+ sage: K_S = project_span(K)
+ sage: J_T1 = project_span(K_S.dual()).dual()
+ sage: J_T2 = project_span(K, K_S.dual())
+ sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice())
+ sage: J_T1 == J_T2
+ True
+ sage: J_T = J_T1
+ sage: l = K.linear_subspace().dimension()
+ sage: codim = K.lattice_dim() - K.dim()
+ sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
+ sage: actual == expected
+ True
+
"""
return len(LL(K))
projects / sage.d.git / blobdiff